Structured Markov chains solver: software tools D. A. Bini, B. Meini, S. Steffe´
∗
Dipartimento di Matematica Universita` di Pisa, Pisa, Italy
bini, meini, steffe @dm.unipi.it
ABSTRACT The package SMC-Solver for solving structured Markov chains is presented. It contains the most advanced algorithms for solving QBD, M/G/1 and G/M/1 problems. The package is provided in two versions: a Matlab toolbox and a Fortran 95 version with a user-friendly graphical interface.
1.
INTRODUCTION
The package Structured Markov Chains Solver (SMC-Solver) for solving the most important classes of Markov chains encountered in queueing models is presented. This package implements the most advanced available algorithms for QBD, M/G/1 and G/M/1 problems. The main features of these algorithms are reported in the paper [2] to which we refer the reader for more details. The algorithms have been implemented in two different software tools: a matlab Toolbox where each method and a set of auxiliary algorithms have been implemented as Matlab functions, and a Fortran 95 package where a set of subroutines are given as a module. A user-friendly graphical interface which allows an easy use of the software tool is provided. The interface is written in C and relies on the GTK graphic libraries. In this paper we describe the main features and the use of these two software tools, and report the results of a set of numerical experiments. The software tools are available upon request from the authors and will be made available online. The main problems solved by our software tools are described below.
1.1 QBD problems 1.1.1 Computing G, R and U ∗This work has been partially supported by MIUR grant number 2004015437. †B. Van Houdt is a postdoctoral fellow of the FWOFlanders.
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B. Van Houdt
†
Department of Mathematics and Computer Science University of Antwerp, Antwerpen, Belgium
[email protected]
Given nonnegative matrices A−1 , A0 , A1 such that A−1 + A0 + A1 is stochastic and irreducible, compute the minimal nonnegative solutions G, R and U of the equations G = A−1 + A0 G + A1 G2 ,
R = A1 + RA0 + R2 A−1 , U = A0 + A1 (I − U )
−1
(1)
A−1 .
1.1.2 Computing π Given in addition the nonnegative matrices B0 , B−1 such that B−1 +A0 +A1 and B0 +A1 are stochastic, if the QBD is positive recurrent, compute the steady state vector π of the QBD defined by A−1 , A0 , A1 and B−1 , B0 , see [2, Section 2.1].
1.2 M/G/1-type Markov chains 1.2.1 Computing G Given P the nonnegative matrices Ai , i = −1, 0, 1, . . ., such that i Ai is stochastic and irreducible, compute the minimal nonnegative solution G of the equation G=
+∞ X
Ai Gi+1 .
(2)
i=−1
In the applications, the infinite sequence of data Ai , i = −1, PK0, 1, 2, . . ., is truncated to the finite size K such that i=−1 Ai is numerically stochastic. In this way, the equation (2) turns into G=
K X
Ai Gi+1 .
(3)
i=−1
1.2.2 Computing π
Given inPaddition the nonnegative matrices B0 , B1 , . . . such that i Bi is stochastic, if the Markov chain is positive recurrent, compute the steady state vector π associated with Ai and Bi , see [2, Section 2.2].
1.3 G/M/1-type Markov chains 1.3.1 Computing R Given P the nonnegative matrices Ai , i = −1, 0, 1, . . ., such that i Ai is stochastic and irreducible, compute the minimal nonnegative solution R of the equation R=
+∞ X
i=−1
Ri+1 Ai .
(4)
1.3.2 Computing π
2.1.1 MaxNumIt and MaxNumComp
Given in addition the nonnegative matrices B0 , B1 , . . ., if the Markov chain is positive recurrent, compute the steady state vector π associated with Ai and Bi , see [2, Section 2.3].
1.4 Non-Skip-Free Markov chains Given the P nonnegative matrices Ai , i = −N, −N + 1, . . ., such that i Ai is stochastic and irreducible, compute the minimal nonnegative solution G of the equation G=
+∞ X
i=−1
Ai G i+1 .
(5)
where Ai are the matrices defined in [2, Section 2.4]. The problem of determining the steady state vector π if the Markov chain is positive recurrent, is also addressed.
2.
THE MATLAB TOOL
The MATLAB implementation of the tool consists of a collection of MATLAB functions which can be executed from the command line (or called from within other functions or scripts). This collection can be partitioned in two subsets: (a) the functions related to Quasi Birth-and-Death (QBD) Markov chains and (b) all functions used to solve M/G/1, G/M/1 and Non-Skip-Free type Markov Chains. Each function takes at least one required parameter as input and may support several optional parameters. A call to a function, called fname, uses the following syntax: output para = fname(required para,optional para) If the function produces a single output parameter O1, one simply replaces output para by O1. In case of multiple outputs, O1, O2, . . . , Ok, one sets output para equal to [O1,O2,. . . ,Ok]. If the user is only interested in the first l < k output variables, it suffices to shorten the list to Ol. As with any other MATLAB function, the required para field holds the list of required parameters separated by commas: R1,R2,. . . ,Rr. The optional para field contains the list of optional parameters. When a function call uses no options, one simple passes the required para field to the function. Otherwise, a pair of inputs must be given for each optional parameter used: the parameter name (pname) and the parameter value (pvalue). The name is always placed between single quotes, the value is placed between quotes if it holds a string (and not a numeric value). The parameter name has to be followed by its value, the order of the optional parameters on the other hand is arbitrary. Hence, in case of t optional parameters we have optional para = ’pname1’,pvalue1,. . .,’pnamet’,pvaluet The tool parses all optional parameters using the support function ParseOptPara.m. As with any inbuilt MATLAB function, help can be requested for a function part of this tool by typing ‘help fname’ on the command line. Before discussing the various functions of the MATLAB tool, we introduce some optional parameters that are supported by most functions.
2.1 General Optional Parameters
All functions that compute the G and/or R matrix are of an iterative nature, except for the Invariant Subspace approach that relies on a Schur decomposition. To avoid infinite loops, a default maximum number of iterations is present for all such functions. The default value is typically about 50 for the algorithms with quadratic convergence and 10000 for those with linear convergence. One can adapt this maximum through the optional parameter MaxNumIt. Its parameter value is simply the maximum number of iterations allowed. Whenever this maximum is reached before satisfying the stopping criteria, a warning is generated. For the functions that compute the components (π 0 , π1 , . . .) of some steady state vector π, the number of components computed is determined dynamically such that the sum of the returned components is at least 1 − 10−9 . To avoid loops, there is a default maximum of 500 components, which can be altered through the optional parameter MaxNumComp.
2.1.2 Verbose To get an idea of the final accuracy and progress of a function call, one can set the Verbose option to 1. For the iterative algorithms, an intermediate check value is printed at the end of each iteration, together with the final residual error of the G and/or R matrix computed. As the number of iterations of the functional iteration (FI) approach is often quite high, one can also assign an integer parameter value k > 1 to the Verbose parameter. This will cause a printout of the check value every k iterations. The default value of this optional parameter is 0, indicating that no feedback is provided. For the functions that determine the steady state probabilities, setting Verbose to 1, causes MATLAB to print the accumulated probability mass of the components computed so far.
2.1.3 Mode A single MATLAB function exist for each family of algorithms that computes the G and/or R matrix. Examples of families are Cyclic Reduction, Functional Iterations, Invariant Subspace approach, etc. Within each family one can still choose among different approaches/algorithms. The optional parameter Mode determines the family member used to perform the computations. A default member was selected among each family. The specific modes for each of the functions concerned is discussed further on.
2.2 Quasi Birth-and-Death functions: computing G, R and U The tool contains five functions to compute the G (R and U ) matrix of a QBD Markov chain. Each of these functions takes three required input parameters: the matrices A−1 , A0 , A1 defined in Section 1.1. Here and hereafter, the three input matrices A−1 , A0 , A1 are denoted by the variables A0, A1 and A2, respectively, so that A0 is the matrix appearing below the main diagonal and A2 above it. Thus, G is the solution to G = A0+ A1G+ A2G2 . The output para field equals [G, R, U], meaning the user can either request the G, the G and R matrix or all three matrices as output variables. Each of these functions first parses the input and optional parameters by calling the QBD ParsePara.m and ParseOptPara.m function. Afterward, a check is performed to see whether one of the matrices A0 or A2 is of rank one, allowing us to provide an explicit solution for G, this is real-
ized by means of the function QBD EG.m, which we discuss in more detail further on. Provided that no explicit solution is available, the actual computation of G starts via the algorithm of choice.
2.2.1 QBD CR.m and QBD LR.m The Cyclic and Logarithmic Reduction (CR and LR) converge quadratically and both support two modes of operation: the Basic and the Shift mode. The difference between both modes exists in the fact that the Shift mode performs some pre- and postprocessing. More specifically, with the ˜ A1 ˜ and A2 ˜ by pershift we first compute the matrices A0, forming the shift operation on A0, A1 and A2. Next, we ˜ A1 ˜ and A2, ˜ instead apply the CR/LR algorithm to A0, of A0, A1 and A2. The postprocessing occurs only in the positive recurrent case and exists in performing a rank one correction to G. The Shift mode is the default mode as it may potentially speed-up the convergence of the CR/LR algorithm. The additional cost of pre- and postprocessing is minor compared to the actual CR/LR execution time. The shift also makes the convergence quadratic in the null recurrent case. The shift operation can be turned off by setting the Mode option to Basic. Both functions support the MaxNumIt and Verbose option.
2.2.2 QBD FI.m The function based on Functional Iterations (FI) achieves linear convergence and has six modes of operation. The first three are the U-Based, Natural and Traditional iteration. The U-Based mode is the default mode as it outperforms the other two in terms of the number of iterations. Three more modes—ShiftU-Based, ShiftNatural and ShiftTraditional—are obtained by combining each scheme with the shift operation, to possibly reduce the number of iterations. Apart from the MaxNumIt and Verbose option, this function also supports the StartValue option. This option allows the user to select another initial matrix G0 , used during the first iteration, where the default choice is G0 = 0. A good choice for recurrent chains is to set G0 = I, while for the transient case having G0 = ηI might result in a significant reduction in the number of iterations. The scalar η < 1 is the largerst eigenvalue of G, it can be found as the Caudal characteristic of the Ramaswami dual of the QBD process of interest (for its computation see QBD Caudal.m).
2.2.3 QBD IS.m The Invariant Subspace (IS) approach function has three modes of operation: the Schur, MSignStandard and MSignBalzer. In the Schur mode, the left invariant subspace of the matrix Z is computed using a Schur decomposition. This method, which is set as the default mode, is a direct method that requires no successive iterations. It relies on the builtin MATLAB function ordschur, which is only supported as of MATLAB 7. If an earlier MATLAB release is used, the function automatically switches to the MSignBalzer mode. The other two modes compute the invariant subspace of Z via the Matrix Sign function sign(Z), which we evaluate via a Newton iteration Zk+1 = Zk /2 + Zk−1 /2 (guaranteeing quadratic convergence). The MSignBalzer implements the Balzer acceleration to improve convergence by adapting the weights 1/2 and computes the determinant of Zk for this purpose. We take the maximum of 10−3 and the determinant’s value to avoid the algorithm from stopping prema-
turely. This function does not support the null recurrent case.
2.2.4 QBD NI.m The Newton Iterations (NI) support three modes of operation: Sylvest, Estimat and DirectSum. They differ in the manner in which they solve the generalized Sylvester matrix equation of the form AXB + XC = D at each iteration. The Estimat mode estimates the solution by letting X = D − AXold B + Xold (I − C) and therefore no longer guarantees quadratic convergence. The Sylvest mode, which is the default mode, solves the equation via an Hessenberg algorithm, implemented by the QBD NI Sylvest.m support function. Finally, the DirectSum mode reduces the problem to a (large) linear system of equations by taking the direct sum of both sides, it is mostly effective for small systems.
2.2.5 QBD EG.m This function determines an explicit solution for G (R and U ) in case A0 or A2 is of rank one. For the recurrent case, we have an immediate expression for G if A0 is rank one, otherwise if A2 is of rank one, R can be expressed in terms of its dominant eigenvalue η (which is computed via the QBD Caudal.m function). The transient cases are solved by combining the previous observations with the Ramaswami dual.
2.3 Quasi Birth-and-Death: steady state and support functions Apart from the functions to compute G, the tool also provides a function to compute the steady state probability vector and the Caudal characteristic. Details on these functions are given below. Figure 1 shows the dependency graph of all the MATLAB functions related to the QBD Markov chains. Support functions are presented as white ellipses. Functions that support multiple modes are represented as square boxes holding a single gray ellipse per mode of operation. The default mode has an additional white periphery. An arrow is draw between two functions if the execution of the first requires a call to the second function. If a support function is only needed in a specific mode, the arrow departs from its corresponding ellipse.
2.3.1 QBD pi.m The tool also contains a function QDB pi.m to compute the steady state probability vector π = (π 0 , π 1 , . . .). This function has three required input parameters: the matrices R, B0 and B1. B0 characterizes the transitions from level one to level zero and B1 from level 0 to itself. In fact, the variables B0 and B1 represent the matrices B−1 and B0 , respectively, of Section 1.1.2. The P function returns the first n + 1 components π 0 to πn if m>n πm e < 10−9 (see also Section 2.1.1). More general boundary conditions can be dealt with through the Boundary option. With this option, transitions from level zero to level one can occur according to a matrix B2 (in the default usage we have B2 = A2). Notice, B0 and B2 do not need to be square in general. The parameter value of the Boundary option should be set to the matrix [B2; A1+ R A2].
2.3.2 QBD Caudal.m This function computes the dominant eigenvalue 0 < η < 1 of the R matrix that solves the matrix equation R = A2+R
picted in Figure 2
QBD_FI.m U-Based
ShiftU-Based
Traditional
ShiftTraditional
Natural
ShiftNatural
2.4.1 MG1 CR.m
QBD_CR.m Shift
Basic
QBD_IS.m MSignBalzer Schur MSignStandard
QBD_Caudal.m
Parsing
QBD_LR.m Basic
Shift
QBD_EG.m
QBD_NI.m
QBD_ParsePara.m
ParseOptPara.m
stat.m
DirectSum
Sylvest
Estimat
The Cyclic Reduction (CR) algorithm achieves quadratic convergence and supports two modes of operation: the ShiftPWCR and the PWCR mode. As in the QBD case, the difference between the two modes exists in the pre- and postprocessing performed under the ShiftPWCR mode. The preprocessing is, in this case, taken care of by the support ˜ = function MG1 Shift.m, which computes a new matrix A ˜ A1 ˜ A2 ˜ . . . Amax]. ˜ [A0 The ShiftPWCR mode is the default mode as it may potentially speed-up the convergence of the CR algorithm. The additional cost of pre- and postprocessing is limited compared to the actual CR execution time. The shift operation can be turned off by setting the Mode option to PWCR. The PWCR algorithm itself computes the matrix power series A(n+1) (z) and Aˆ(n+1) (z) (see [2, Section 4.3]) through a point-wise evaluation. The point-wise evaluation is implemented by dynamically doubling the number of roots used in the FFT evaluation. A default maximum of 2048 roots is enforced for each iteration. One can adapt this number using the MaxNumRoot option. Whenever this maximum is attained, we generate a warning that the doubling process was interrupted prematurely. The MG1 CR function also supports the MaxNumIt and Verbose option.
2.4.2 MG1 FI.m
QBD_pi.m
QBD_NI_Sylvest.m
Figure 1: Dependency graph for the Quasi Birthand-Death Functionality A1 + R2 A0, where A0, A1 and A2 characterize a recurrent QBD Markov chain. This dominant eigenvalue is computed via a bisection algorithm and is also known as the Caudal characteristic. The function takes the matrices A0, A1 and A2 as required input parameters. Setting the option Dual to 1 causes QBD Caudal.m to compute the dominant eigenvalue of the Ramaswami dual and must only be used for transient Markov chains. This option is useful for computing an alternate StartValue for the functional iteration (FI) algorithm.
2.4 M/G/1 and G/M/1 functions: computing G and R All functions implementing an algorithm to compute the G or R matrix of an M/G/1 or G/M/1 type Markov chain, respectively, demand a single required input parameter: the matrix A = [A0 A1 A2 . . . Amax], where the variables A0, A1, . . . , Amax denote the matrices A−1 , A0 , A1 , . . . , AK of equation (3). For M/G/1 type Markov chains, A0 is the block appearing below the main diagonal, for the G/M/1 type, A0 appears above the main diagonal. Four functions are supported to compute the G matrix of the M/G/1 problem, while a single function, that relies on these four functions via the Ramaswami dual, is available for the G/M/1 setting. Each of these functions start, via the support function MG1 EG.m, by checking whether G can be determined explicitly, which is the case if A0 is of rank one. The dependency graph for the M/G/1 and G/M/1 functions is de-
The Functional Iteration (FI) approach supports the same six modes of operations and optional parameters as its QBD counterpart (see Section 2.2.2 for details). An additional optional parameter for the M/G/1 setting is the NonZeroBlocks option. This option is mostly useful when the majority of the Ai blocks is equal to zero. Let S = {s1, . . . , sn} be the positive indices of the non-zero blocks of A. Then, setting the required parameter A=[A0 As1 As2 . . . Asn] and the option value of NonZeroBlocks to [s1 s2 . . . sn], may result in a substantial gain in the computation time. It is also useful to avoid unnecessary memory usage of the function call. The current implementation does not support the combination of the NonZeroBlocks option with either one of the Shift modes. If the user does provoke such a call, the shift operation is ignored.
2.4.3 MG1 IS.m The Invariant Subspace (IS) function for M/G/1 type Markov chains has the same modes of operation and restrictions as the QBD IS.m function (see Section 2.2.3). Except that the MSignBalzer is now the default mode, as it tends to be faster than the Schur mode. The current P implementation only deals with the case where A(z) = i Ai z i is a matrix polynomial. The memory usage can be considerable as the dimension of the square companion matrix is a function of the degree of A(z). The IS approach is especially effective when A(z) is a rational power series (when the numerator and denominator have a fairly small degree). We plan to support such cases in a future implementation of the MG1 IS.m function.
2.4.4 MG1 RR.m The Ramaswami Reduction (RR) function converges quadratically and has three modes of operation: Direct, DispStruct and DispStructFFT. Let m be the block size of the original M/G/1 type Markov chain and N be the degree of
MG1_RR_tempBFFT.m
MG1_RR.m
Direct
DispStructFFT
MG1_RR_BtempFFT.m
DispStruct
MG1_RR_tempB.m
MG1_IS.m
MG1_RR_Btemp.m Schur
MSignBalzer
MSignStandard
GIM1_R.m
MG1_CR.m ShiftPWCR
MG1_EG.m
GIM1_Caudal.m
PWCR
ParseOptPara.m MG1_pi.m stat.m
MG1_FI.m U-Based
ShiftU-Based
Traditional
ShiftTraditional
Natural
ShiftNatural
MG1_Shift.m
GIM1_pi.m
Figure 2: Dependency graph for the M/G/1 and G/M/1 Functionality the power series A(z). The Direct mode ignores the displace(n) ment structure of the matrices A1 (see [2, Section 4.4]) and simply stores them entirely. The time and memory complexity per iteration therefore equals O((mN )3 ) and O((mN )2 ), (n) respectively. Both other modes exploit the fact that A1 can be expressed in terms of five block lower triangular block Toeplitz matrices (thereby reducing the memory usage to O(m2 N )). The DispStructFFT mode uses FFTs to com(n) pute all products between A1 and a block column vector, whereas the DispStruct mode uses a standard approach. The resulting time complexity equals O(m2 N ∗ log N ∗ + m3 N ∗ ), where N ∗ is the smallers power of 2 such that N ∗ ≥ N , with the FFTs and O(m3 N 2 ) without them. Both the DispStruct and DispStructFFT make use of two MATLAB support scripts (see Figure 2). For small systems, the Direct mode is superior and therefore set as the default mode. For larger systems the choice between the other two modes depends on the values of m and N .
2.4.5 GIM1 R.m This function computes the R matrix of a G/M/1 type Markov chain. It first transforms the G/M/1 type problem characterized by the required input matrix A, into a M/G/1 ` of A. Afterward, problem by taking the Ramaswami dual A ` to one of the M/G/1 functions to compute the it passes A G matrix of the dual. The second required input parameter Algor specifies the algorithm of choice, it is one of the strings CR, FI, IS or RR. Any optional parameters, including the NonZeroBlocks option of the FI, for the M/G/1 call can be passed as optional parameters to the GIM1 R.m function.
The R matrix of interest is retrieved as the dual of G.
2.5 M/G/1 and G/M/1: steady state and support functions 2.5.1 MG1 pi.m By calling this function, one computes the steady state vector of a positive recurrent M/G/1 type Markov chain (see Section 2.1.1 for info on the number of components computed). There are three required input parameters: B = [B0 B1 B2 . . . Bbmax], A = [A0 A1 A2 . . . Aamax] and the matrix G. The matrix Bi characterizes the transitions from level 0 to level i. For the specific case where B = A, one simply assigns the empty matrix [ ] to B, i.e., MG1 pi([ ],A,G). In its default usage, transitions from level 1 to 0 are assumed to occur according to the matrix A0. The more general case, where C0 covers these transitions, can be treated via the Boundary option. The parameter value of this option must contain the matrix C0. The components of π are computed via Ramaswami’s formula. An implementation of the fast version of Ramaswami’s formula, which is mainly effective if a very large number of components is required, is currently not supported. Finally, one can also pass the first component π 0 as input to the MG1 pi.m function by setting the value of the InputPiZero option equal to π 0 . This option is used by the NSF pi.m function.
2.5.2 GIM1 pi.m The stationary vector of a positive recurrent G/M/1 type Markov chain is obtained through this function (see Section
2.1.1 for info on the number of components computed). It takes two required input parameters: B = [B1; B2; B3; . . . ; Bbmax], where Bi governs the transitions from level i − 1 to level 0, i.e., it is the matrix B−i+1 appearing in [2, Section 2.3], and the matrix R. For the more general case where the transitions from level 0 to 1 are described by a matrix B0 6= A0, one can evoke the Boundary option, the parameter value of which has to equal [B0; A1; A2; . . . ; Aamax].
face. The entire package, which is still work in progress, is available from the authors upon request.
2.6 Non-Skip-Free Markov chain functions
subroutine crqbd(An1, eps, maxiter, G, subroutine lrqbd(An1, eps, maxiter, G,
Non-Skip-Free (NSF) type Markov chains can be solved in two manners. The first exists in reblocking the system to an M/G/1 type Markov chain and applying the standard algorithms for the M/G/1 type paradigm (i.e., the new block size equals N m, where N was the number of blocks below the main diagonal and m the original block size). Alternatively, when computing G, one could also exploit the specific structure of the blocks as explained in [2, Section 4.1]. The tool contains two functions implementing the latter approach: NSF GHT.m and NSF pi.m which are discussed below.
2.6.1 NSF GHT.m This function computes the mN × mN size G matrix using the functional iterations developed by Gail, Hantler and Taylor (GHT), which converges linearly. It has twoe required input parameters the matrix A = [A0 . . . Amax] and an integer value N , where A0 appears N blocks below the main diagonal and AN on the main diagonal. Thus, the variable Ai holds the matrix Ai−N of [2, Section 2.4]. The GHT algorithm first computes the first block row of G, which fully characterizes G, in an iterative manner and subsequently constructs the remaining block rows in a direct manner. Apart from the Verbose and MaxNumIt options, one can also set the FirstBlockRow option value to one. As a result only the first block row is returned as an output variable (which can useful to suppress the memory usage). The default value setting for FirstBlockRow is zero.
2.6.2 NSF pi.m The steady state vector of the NSF Markov chain can be computed through this function. It has three required input parameters: B, A and G. The B matrix holds the first N block rows of the transition matrix P , A equals [A0 . . . Amax] as in the NSF GHT.m function and G is the corresponding G matrix of size mN . Ramaswami’s formula is used to generate the components of the π vector (the number of which can be affected by the MaxNumComp option). For NSF chains where each of the first N block rows is identical to A, one simply sets B equal to the empty matrix [ ]. In such particular cases we rely on a more efficient computation of the first N components of π. When setting the FirstBlockRow option to one, it suffices to pass the first block row of G as input.
3.
FORTRAN PACKAGE
Most part of the algorithms described in the paper [2] have been implemented in a Fortran 95 module. The subroutines can be called by the user inside a main program or, alternatively, can be used in an interactive way through a graphical interface written in C. The graphical interface relies on the GTK library and runs under the linux system. In this section we give an outline about the features and the use of the Fortran subroutines and of the graphical inter-
3.1 The Fortran 95 subroutines We report the main subroutines implemented in the package. The first two subroutines crqbd and lrqbd compute the minimal solutions G, R and U of the equations (1). They have the same syntax, i.e., A0, A1, doshift, R, U, drift, err A0, A1, doshift, R, U, drift, err
dogth,& ) dogth,& )
where An1, A0 and A1 are the matrix variables containing the matrices A−1 , A0 and A1 respectively; the logical variables doshift and dogth, if .true., perform the shift acceleration of [2, Section 3] and the GTH trick of [5] for improving numerical stability. This trick cannot be applied if the shift acceleration is performed. The optional input variables eps and maxiter contain the error bound for the stop condition (by default eps=1.e-12) and the maximum number of allowed iterations (by default maxiter=50), respectively. The optional output variables G, R and U contain the solutions G, R and U , respectively; drift contains the value of the drift µ; while error is the residual error. The subroutines pwcr and spwcr compute the solution of the equation (3) by means of the algorithm of point-wise cyclic reduction without the shift acceleration (pwcr) and with the shift acceleration (spwcr). Their syntax is the same, i.e., subroutine pwcr(A, eps, maxiter, & maxintp, G, drift, err) subroutine spwcr(A, eps, maxiter, & maxintp, G, drift, err) In both the subroutines, A is a three-way array where A(:,:,k) contains the block Ak−2 for k = 1, 2, . . . , K + 2; the optional input variables eps, maxiter and maxintp contain an error bound used in the stop condition (by default eps=1e-12), the maximum number of iterations (by default maxiter=50) and the maximum number of interpolation points (by default maxintp=256), respectively. The output variables G, drift and err contain the solution G of the matrix equation (3), the drift µ and the residual error, respectively. The subroutine fi computes G by means of functional iterations. The syntax is subroutine fi(A, doshift, method, eps, & maxiter, x0, G, drift, err) Here, A is a three-way array where A(:,:,k) contains the block Ak−2 for k = 1, 2, . . . , K + 2; doshift is an optional logical variable, if .true. it performs the shift technique (default doshift=.false.); method is an optional integer variable which selects the functional iteration: 1→Natural iteration, 2→Traditional iteration, 3→Method based on the matrix U (default method=3); the optional input variables eps and maxiter contain an error bound on the stop condition (by default eps=1e-12) and the maximum number of iterations (by default maxiter=10000); X0 is an optional input variable which contains the initial approximation (by default X0=0 if doshift=.true., X0=rI if doshift=.false., where r=ρ(G)). The output variables G, drift and err contain the solution G of the matrix equation (3), the drift µ and the residual error, respectively. The subroutine ni computes G by means of Newton’s it-
eration. The syntax is subroutine ni(A, eps, maxiter, G, & drift, err) Here, A is a three-way array where A(:,:,k) contains the block Ak−2 for k = 1, 2, . . . , K + 2; the optional input variables eps and maxiter contain an error bound used in the stop condition (by default eps=1e-12) and the maximum number of iterations (by default maxiter=50). The output variables G, drift and err contain the solution G of the matrix equation (3), the drift µ and the residual error, respectively. The subroutine is implements the invariant subspace method. Its syntax is subroutine is(A, method, eps, maxiter, & G, drift, err) Here, A is a three-way array where A(:,:,k) contains the block Ak−2 for k = 1, 2, . . . , K + 2; method is an optional integer variable which selects the algorithm for computing the invariant subspace: 1→ Matrix Sign Iteration; 2→ Matrix Sign Iteration with the Balzer acceleration; 3→ Schur Decomposition; by default method=2. The optional input variables eps and maxiter contain an error bound on the stop condition (by default eps=1e-12) and the maximum number of iterations (by default maxiter=50). The output variables G, drift and err contain the solution G of the matrix equation (3), the drift µ and the residual error, respectively.
main window. The level of verbosity is selectable. The buffer with the log of the session can be saved in a file. The computed solutions (matrices, vectors) can be saved in several ASCII formats. Morover a sparse matrix format is suppported for reading or writing data (the computations however do not use the sparse matrix format internally). A viewing and editing graphical tool allows to display and edit input matrices and to view and compare output matrices. This is performed in a graphical way by representing each element of the matrices as a small gray (or colored) square, where the intensity varies according to the magnitude of the number. A good choice of the color scale allows an immediate and intuitive viewing of some structure properties of the matrix. The numerical values of the entries are displayed and edited in a separate small window as the user moves the cursor on the colored squares. A simple online window help is available. In Figures 3, 4, 5 we display some examples of the graphical interface.
3.2 The graphical interface We have implemented a simple Graphical Interface in C relying on the GTK+ 2.0 graphical libraries. The executable runs in Linux Boxes where the corresponding GTK2 shared libraries are available. The Fortran routines are linked statically, so that the executable needs neither Fortran Compiler nor Fortran run-time libraries to run. A fully statically linked program is discouraged by GTK+ developers. By means of menus in the main window, the user can easily choose the kind of problem (QBD, M/G/1, G/M/1), either by loading one of the examples provided in the package, or by loading the input data from some external ASCII files. In the first case, an example window allows the user to to set up the size of the problem and some parameters. Several formats are supported for the input files. From the main menu the user can choose the desired algorithm among Cyclic Reduction, Logarithmic Reduction, Functional Iterations, Invariant Subspace. For each algorithm the user can set additional optional parameters like the shift acceleration, the diagonal adjustment, or can modify the values of the maximum number of iterations, the number of interpolations points in cyclic reduction, the error bound for the stop condition. For each algorithm, the user can select the goals, for instance, computing G, R, U or π. Warnings are flashed in the status line of the window for inconsistent choices of the user and also if the selected algorithms or options are not yet implemented. The use of multithread support enables the user to start/stop a computation just by pressing a button. During the numerical computation, partial results, timings, and information on the evolution of the computation are displayed in real time on a scrollable viewport of the
Figure 3: Graphical User Interface: main window
Figure 4: Graphical User Interface: main window
4. NUMERICAL EXAMPLES Various numerical examples performed on some test problems are presented in this section. Since the Matlab and the Fortran versions implement the same algorithms, they perform similarly concerning the number of iterations and the residual error. For this reason, we report only the results of the Matlab version.
Figure 5: Graphical User Interface: displaying G
process of the corresponding queueing system) in the latter two examples, while for the DELTA case it is below 0.01. Whenever η2 is not too close to one (in the positive recurrent case), some reduction may be expected by performing the shift. For the cases where the shift does not result in a reduction, only a limited amount of resources is waisted by performing it. Another observation related to the shift is that setting the StartValue equal to the identity matrix I seems to have a similar effect as the shift operation when applied to the FI approach. When comparing the CR and LR algorithm, we find that both perform very similar, except that the CR algorithm is slightly faster. We also observe that among the different IS modes, the Schur approach results in the shortest computation times in all examples. Although the NI algorithm often requires the lowest number of iterations, it tends to be considerably slower. Recall, the memory requirement of the DirectSum approach is O(m4 ), with m the block size; therefore, we did not apply it to the TANDEM example where m = 388.
4.2 M/G/1 type Markov chains 4.1 QBD problems In this section we present some numerical examples to demonstrate the performance of the QBD MATLAB functions. This section also assists the user in making his/her choice between the different algorithms available. Of all tests conducted we report on four examples. It concerns (i) a QBD used to analyze a wireless MAC protocol named FS-ALOHA [4], (ii) the 32 × 32 QBD that was previously studied in [1] (with δ = 10−3 ), (iii) a chain with blocks of size 40 introduced in [6] to study the behavior of a video playout buffer and (iv) the QBD considered in [8] to analyze a discrete time tandem queue, it has blocks of size 388. We refer to these examples as FS-ALOHA, DELTA, VIDEOPLAY and TANDEM. The Markov chain for each of these examples is positive recurrent. Timing was performed by MATLAB on an Intel 2Ghz Pentium with 512 kB RAM in Realtime mode under Windows. As the MATLAB timing granularity on such a system is not very fine (in the millisecond range), most computations were averaged over several runs. Whenever the maximum number of iterations was reached, we only reported the computation time and residual error up to the maximum number of allowed iterations (which was lowered to 1000 for the TANDEM case, due to the large block size). For the FS-ALOHA case, A0 is of rank one, meaning that an explicit expression for G and R was available. All functions detected the rank one property through the QBD EG.m function. Therefore, all algorithms required the same computation time without any iterations. Table 1 reports the results when computing the G, R and U matrix, in each of the remaining three examples. The values in the column labeled (R) hold the order of the maximum residual error (for the three matrices), e.g., having a residue of 2.5 10−12 implies that (R) equals −12. Let us briefly discuss a number of key observations. The Shift operation performed by the CR, LR and FI algorithm results in a strong decrease in the number of iterations in the DELTA example, whereas for the other two examples, there is either no or hardly any gain. This can be explained by the fact that the second largest eigenvalue η2 of G is very close to one (above 0.99, which seems to be caused by the strong correlation in the arrival
In this section we present a similar comparison, but for the M/G/1 type paradigm. Three examples where considered that each solve a D-BMAP/D/1 queue by means of an M/G/1 type Markov chain with a different arrival process, i.e., D-BMAP. In the first two examples the D-BMAP is a superposition of M ON/OFF sources as introduced in [3]. Both the ON and OFF period are geometric with a mean of 1/a slots. While on, a source generates a packet with probability 1/d. In the first example, which we refer to as VBR – 20, we consider M = 20 sources, with a mean ON period of 100 slots and let d = 10.5, meaning the load of the queueing system equals .9524. In the second case, we obtain a transient chain by setting M = 5, a = .0001 and d = 1.1, meaning the load is very high and equals 4.5454. Finally, example three is the D-BMAP developed in [7] to approximate a superposition of various D-BMAPs, each used to model a variable bit rate source, by a circulant D-BMAP. In the first two examples both the block size m and the number of nonzero blocks N equals M + 1, while in the third we have N = 25 blocks each of size m = 6. An overview of the results is shown in Table 2. We can see that the Shift is quite effective in the first and third example, as η2 = .9170 and −0.3824, respectively. For the VBR – 5 example we have η1 = .9996 and η2 = .9946. We also notice that even though the Shift might realize a reduction in the number of iterations of the CR algorithm, its computation time may be larger. This is caused by the higher number of roots required at each iteration to perform the point-wise evaluation. For all three examples, the fastest mode for the RR algorithm is the Direct mode. The other two modes become more effective when N becomes large (recall, the memory complexity of the Direct mode is a square function of N ). For the IS approach, we find that the residual error for the Schur mode is acceptable for the VBR – 5 example, where there are only 6 nonzero blocks, which is no longer the case for the other examples. Additional experiments indicate that the IS method manages to produce a small residual error if the degree of A(z) is very limited. The other two IS modes of operation failed as, due to numerical errors, the dimension of the left invariant subspace did not match m (generating an error).
Example
DELTA #It Time Cyclic Reduction (CR) Shift 3 .0084 Basic 14 .0217 Logarithmic Reduction (LR) Shift 3 .0092 Basic 14 .0255 Invariant Subspace (IS) Schur 1 .0344 MSignBalzer 14 .0567 MSignStandard 15 .0537 Newton Iterarions (NI) Sylvest 14 .2364 Estimat 6327 7.234 DirectSum 14 50.67 Functional Iterarions (FI) U-Based 6889 5.313 ShiftU-Based 5 .0092 U-Based + Start 5 .0078 Natural 10000+ 3.297 ShiftNatural 8 .0077 Natural + Start 9 .0072 Tradit. 10000+ 4.14 ShiftTradit. 5 .0080 Tradit. + Start 6 .0078
(R)
VIDEOPLAY #It Time (R)
#It
TANDEM Time (R)
-16 -16
10 10
.0269 .0242
-15 -15
12 12
13.16 12.80
-14 -15
-16 -16
10 10
.0297 .0283
-15 -15
12 12
16.96 16.59
-14 -15
-13 -12 -11
1 13 14
.0236 .0708 .0758
-15 -13 -13
1 9 16
24.08 33.39 29.24
-13 -12 -14
-16 -13 -16
7 450 7
.2634 .75 84.37
-15 -13 -15
9 1000+ –
278.6 899.1 –
-15 -12 –
-16 -16 -16 -10 -16 -16 -13 -16 -16
526 481 450 1174 1075 1004 653 481 561
.5541 .5106 .5453 .5059 .4869 .4666 .4459 .3334 .3928
-15 -15 -15 -14 -14 -14 -15 -15 -15
1000+ 1000+ 1000+ 1000+ 1000+ 1000+ 1000+ 1000+ 1000+
431.8 432.1 431.5 215.9 216.2 586.3 404.6 405.0 815.8
-7 -8 -9 -7 -7 -8 -7 -1 -9
Table 1: Timing Results QBDs
Example
VBR – 20 #It Time Cyclic Reduction (CR) Shift 11 5.242 Basic 10 1.578 Ramaswami Reduction (RR) Direct 11 1.922 DispStruct 11 9.406 DispStructFFT 12 26.85 Invariant Subspace (IS) Schur 1 3.555 Functional Iterarions (FI) U-Based 1147 1.101 ShiftU-Based 224 .2420 U-Based + Start 206 .2190 Natural 2560 1.750 ShiftNatural 525 .3750 Natural + Start 481 .3435 Tradit. 1635 1.640 ShiftTradit. 247 .2500 Tradit. + Start 303 .3125
VBR – 5 Time
(R)
CIRCULANT #It Time (R)
18 17
3.438 3.928
-16 -16
7 10
.3562 .1616
-16 -14
-16 -16 -16
17 17 17
.0084 .2000 1.194
-16 -16 -16
12 12 12
.1256 .7384 2.155
-16 -16 -16
-7
1
.0094
-13
1
.1706
-5
-15 -15 -15 -15 -15 -15 -15 -15 -15
10000+ 10000+ 4503 10000+ 10000+ 5431 10000+ 10000+ 4553
1.450 1.462 .6594 .5844 .5906 .3186 1.656 1.637 .7406
-7 -9 -15 -7 -9 -15 -7 -9 -15
1119 19 20 2270 50 51 1740 16 36
.3287 .0084 .0075 .4612 .0131 .0125 .5391 .0075 .0134
-15 -16 -16 -15 -15 -15 -15 -15 -15
(R)
#It
-16 -15
Table 2: Timing Results M/G/1 type
5.
REFERENCES
[1] N. Akar and K. Sohraby. An invariant subspace approach in M/G/1 and GI/M/1 type Markov chains. Stochastic Models, 13(3):381–416, 1997. [2] D. A. Bini, B. Meini, S. Steff´e, and B. Van Houdt. Structured Markov chains solver: algorithms. Proceedings of SMCTOOLS, Pisa 2006, pages –, 2006. [3] C. Blondia. A discrete-time batch Markovian arrival process as B-ISDN traffic model. Belgian Journal of Operations Research, Statistics and Computer Science, 32(3,4):3–23, 1993. [4] D. V. Cortizo, J. Garc´ıa, C. Blondia, and B. Van Houdt. FIFO by sets ALOHA (FS-ALOHA): a collision resolution algorithm for the contention channel in wireless ATM systems. Performance Evaluation, 36-37:401–427, 1999. [5] W. K. Grassmann, M. I. Taksar, and D. P. Heyman. Regenerative analysis and steady state distributions for Markov chains. Oper. Res., 33(5):1107–1116, 1985. [6] T. Hofkens, K. Spaey, and C. Blondia. Transient analysis of the D-BMAP/G/1 queue with an application to the dimensioning of a playout buffer for VBR traffic. In Proc. of Networking 2004, Athens, Greece, 2004. [7] K. Spaey and C. Blondia. Circulant matching method for multiplexing ATM traffic applied to video sources. In Proc. Perf. of Infor. and Comm. Systems (PICS), pages 234–245, Lund (Sweden), 1998. [8] B. Van Houdt and A. Alfa. The response time in a discrete time tandem queue with blocking, markovian arrivals and phase-type services. Operations Research Letters, 33:4, 2005.
